VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK.DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l’environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France.ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal.JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal.MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK.TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK.MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA.MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland.SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.

"This book presents a comprehensive approach to the theory of extreme events (extreme values) from the perspective of dynamical systems. It is nicely written by a group of authors from different parts of the globe".... "This book presents all the signs of becoming a significant tool in pure and applied mathematical research" Idris Assani on behalf of Mathematical Reviews, October 17

• 1 Introduction 11.1 A Transdisciplinary Research Area 11.2 Some Mathematical Ideas 41.3 Some Difficulties and Challenges in Studying Extremes 61.3.1 Finiteness of Data 61.3.2 Correlation and Clustering 81.3.3 Time Modulations and Noise 91.4 Extremes Observables and Dynamics 101.5 This Book 12Acknowledgments 142 A Framework for Rare Events in Stochastic Processes and Dynamical Systems 172.1 Introducing Rare Events 172.2 Extremal Order Statistics 192.3 Extremes and Dynamics 203 Classical Extreme Value Theory 233.1 The i.i.d. Setting and the Classical Results 243.1.1 Block Maxima and the Generalized Extreme Value Distribution 243.1.2 Examples 263.1.3 Peaks Over Threshold and the Generalized Pareto Distribution 283.2 Stationary Sequences and Dependence Conditions 293.2.1 The Blocking Argument 303.2.2 The Appearance of Clusters of Exceedances 313.3 Convergence of Point Processes of Rare Events 323.3.1 Definitions and Notation 333.3.2 Absence of Clusters 353.3.3 Presence of Clusters 353.4 Elements of Declustering 374 Emergence of Extreme Value Laws for Dynamical Systems 394.1 Extremes for General Stationary Processes—an Upgrade Motivated by Dynamics 404.1.1 Notation 414.1.2 The New Conditions 424.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 444.1.4 Proofs of Theorem 4.1.4 and Corollary 4.1.5 464.2 Extreme Values for Dynamically Defined Stochastic Processes 514.2.1 Observables and Corresponding Extreme Value Laws 534.2.2 Extreme Value Laws for Uniformly Expanding Systems 574.2.3 Example Revisited 594.2.4 Proof of the Dichotomy for Uniformly Expanding Maps 614.3 Point Processes of Rare Events 624.3.1 Absence of Clustering 624.3.2 Presence of Clustering 634.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes 654.4 Conditions Дq(un), D3(un) Dp(un)∗ and Decay of Correlations 664.5 Specific Dynamical Systems Where the Dichotomy Applies 704.5.1 Rychlik Systems 704.5.2 Piecewise Expanding Maps in Higher Dimensions 714.6 Extreme Value Laws for Physical Observables 725 Hitting and Return Time Statistics 755.1 Introduction to Hitting and Return Time Statistics 755.1.1 Definition of Hitting and Return Time Statistics 765.2 HTS Versus RTS and Possible Limit Laws 775.3 The Link Between Hitting Times and Extreme Values 785.4 Uniformly Hyperbolic Systems 845.4.1 Gibbs Measures 855.4.2 First HTS Theorem 865.4.3 Markov Partitions 865.4.4 Two-Sided Shifts 885.4.5 Hyperbolic Diffeomorphisms 895.4.6 Additional Uniformly Hyperbolic Examples 905.5 Nonuniformly Hyperbolic Systems 915.5.1 Induced System 915.5.2 Intermittent Maps 925.5.3 Interval Maps with Critical Points 935.5.4 Higher Dimensional Examples of Nonuniform Hyperbolic Systems 945.6 Nonexponential Laws 956 Extreme Value Theory for Selected Dynamical Systems 976.1 Rare Events and Dynamical Systems 976.2 Introduction and Background on Extremes in Dynamical Systems 986.3 The Blocking Argument for Nonuniformly Expanding Systems 996.3.1 Assumptions on the Invariant Measure μ 996.3.2 Dynamical Assumptions on (f, χ, μ) 996.3.3 Assumption on the Observable Type 1006.3.4 Statement or Results 1016.3.5 The Blocking Argument in One Dimension 1026.3.6 Quantification of the Error Rates 1026.3.7 Proof of Theorem 6.3.1 1076.4 Nonuniformly Expanding Dynamical Systems 1086.4.1 Uniformly Expanding Maps 1086.4.2 Nonuniformly Expanding Quadratic Maps 1096.4.3 One-Dimensional Lorenz Maps 1106.4.4 Nonuniformly Expanding Intermittency Maps 1106.5 Nonuniformly Hyperbolic Systems 1136.5.1 Proof of Theorem 6.5.1 1156.6 Hyperbolic Dynamical Systems 1166.6.1 Arnold Cat Map 1166.6.2 Lozi-Like Maps 1186.6.3 Sinai Dispersing Billiards 1196.6.4 Hénon Maps 1196.7 Skew-Product Extensions of Dynamical Systems 1206.8 On the Rate of Convergence to an Extreme Value Distribution 1216.8.1 Error Rates for Specific Dynamical Systems 1236.9 Extreme Value Theory for Deterministic Flows 1266.9.1 Lifting to Xh 1296.9.2 The Normalization Constants 1296.9.3 The Lap Number 1306.9.4 Proof of Theorem 6.9.1 1316.10 Physical Observables and Extreme Value Theory 1336.10.1 Arnold Cat Map 1336.10.2 Uniformly Hyperbolic Attractors: The Solenoid Map 1376.11 Nonuniformly Hyperbolic Examples: the HÉNON and LOZI Maps 1406.12 Extreme Value Statistics for the Lorenz ’63 Model 1417 Extreme Value Theory for Randomly Perturbed Dynamical Systems 1457.1 Introduction 1457.2 Random Transformations via the Probabilistic Approach: Additive Noise 1467.2.1 Main Results 1497.3 Random Transformations via the Spectral Approach 1557.4 Random Transformations via the Probabilistic Approach: Randomly Applied Stochastic Perturbations 1597.5 Observational Noise 1637.6 Nonstationarity—the Sequential Case 1658 A Statistical Mechanical Point of View 1678.1 Choosing a Mathematical Framework 1678.2 Generalized Pareto Distributions for Observables of Dynamical Systems 1688.2.1 Distance Observables 1698.2.2 Physical Observables 1728.2.3 Derivation of the Generalized Pareto Distribution Parameters for the Extremes of a Physical Observable 1748.2.4 Comments 1768.2.5 Partial Dimensions along the Stable and Unstable Directions of the Flow 1778.2.6 Expressing the Shape Parameter in Terms of the GPD Moments and of the Invariant Measure of the System 1788.3 Impacts of Perturbations: Response Theory for Extremes 1808.3.1 Sensitivity of the Shape Parameter as Determined by the Changes in the Moments 1828.3.2 Sensitivity of the Shape Parameter as Determined by the Modification of the Geometry 1858.4 Remarks on the Geometry and the Symmetries of the Problem 1889 Extremes as Dynamical and Geometrical Indicators 1899.1 The Block Maxima Approach 1909.1.1 Extreme Value Laws and the Geometry of the Attractor 1919.1.2 Computation of the Normalizing Sequences 1929.1.3 Inference Procedures for the Block Maxima Approach 1949.2 The Peaks Over Threshold Approach 1969.2.1 Inference Procedures for the Peaks Over Threshold Approach 1969.3 Numerical Experiments: Maps Having Lebesgue Invariant Measure 1979.3.1 Maximum Likelihood versus L-Moment Estimators 2039.3.2 Block Maxima versus Peaks Over Threshold Methods 2049.4 Chaotic Maps With Singular Invariant Measures 2049.4.1 Normalizing Sequences 2059.4.2 Numerical Experiments 2089.5 Analysis of the Distance and Physical Observables for the HNON Map 2129.5.1 Remarks 2189.6 Extremes as Dynamical Indicators 2189.6.1 The Standard Map: Peaks Over Threshold Analysis 2199.6.2 The Standard Map: Block Maxima Analysis 2209.7 Extreme Value Laws for Stochastically Perturbed Systems 2239.7.1 Additive Noise 2259.7.2 Observational Noise 22910 Extremes as Physical Probes 23310.1 Surface Temperature Extremes 23310.1.1 Normal Rare and Extreme Recurrences 23510.1.2 Analysis of the Temperature Records 23510.2 Dynamical Properties of Physical Observables: Extremes at Tipping Points 23810.2.1 Extremes of Energy for the Plane Couette Flow 23910.2.2 Extremes for a Toy Model of Turbulence 24510.3 Concluding Remarks 24711 Conclusions 24911.1 Main Concepts of This Book 24911.2 Extremes Coarse Graining and Parametrizations 25311.3 Extremes of Nonautonomous Dynamical Systems 25511.3.1 A Note on Randomly Perturbed Dynamical Systems 25811.4 Quasi-Disconnected Attractors 26011.5 Clusters and Recurrence of Extremes 26111.6 Toward Spatial Extremes: Coupled Map Lattice Models 262Appendix A Codes 265A.1 Extremal Index 266A.2 Recurrences—Extreme Value Analysis 267A.3 Sample Program 271References 273Index 293

VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK.DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l’environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France.ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal.JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal.MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK.TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK.MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA.MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland.SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.

"This book presents a comprehensive approach to the theory of extreme events (extreme values) from the perspective of dynamical systems. It is nicely written by a group of authors from different parts of the globe".... "This book presents all the signs of becoming a significant tool in pure and applied mathematical research" Idris Assani on behalf of Mathematical Reviews, October 17

• 1 Introduction 11.1 A Transdisciplinary Research Area 11.2 Some Mathematical Ideas 41.3 Some Difficulties and Challenges in Studying Extremes 61.3.1 Finiteness of Data 61.3.2 Correlation and Clustering 81.3.3 Time Modulations and Noise 91.4 Extremes Observables and Dynamics 101.5 This Book 12Acknowledgments 142 A Framework for Rare Events in Stochastic Processes and Dynamical Systems 172.1 Introducing Rare Events 172.2 Extremal Order Statistics 192.3 Extremes and Dynamics 203 Classical Extreme Value Theory 233.1 The i.i.d. Setting and the Classical Results 243.1.1 Block Maxima and the Generalized Extreme Value Distribution 243.1.2 Examples 263.1.3 Peaks Over Threshold and the Generalized Pareto Distribution 283.2 Stationary Sequences and Dependence Conditions 293.2.1 The Blocking Argument 303.2.2 The Appearance of Clusters of Exceedances 313.3 Convergence of Point Processes of Rare Events 323.3.1 Definitions and Notation 333.3.2 Absence of Clusters 353.3.3 Presence of Clusters 353.4 Elements of Declustering 374 Emergence of Extreme Value Laws for Dynamical Systems 394.1 Extremes for General Stationary Processes—an Upgrade Motivated by Dynamics 404.1.1 Notation 414.1.2 The New Conditions 424.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 444.1.4 Proofs of Theorem 4.1.4 and Corollary 4.1.5 464.2 Extreme Values for Dynamically Defined Stochastic Processes 514.2.1 Observables and Corresponding Extreme Value Laws 534.2.2 Extreme Value Laws for Uniformly Expanding Systems 574.2.3 Example Revisited 594.2.4 Proof of the Dichotomy for Uniformly Expanding Maps 614.3 Point Processes of Rare Events 624.3.1 Absence of Clustering 624.3.2 Presence of Clustering 634.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes 654.4 Conditions Дq(un), D3(un) Dp(un)∗ and Decay of Correlations 664.5 Specific Dynamical Systems Where the Dichotomy Applies 704.5.1 Rychlik Systems 704.5.2 Piecewise Expanding Maps in Higher Dimensions 714.6 Extreme Value Laws for Physical Observables 725 Hitting and Return Time Statistics 755.1 Introduction to Hitting and Return Time Statistics 755.1.1 Definition of Hitting and Return Time Statistics 765.2 HTS Versus RTS and Possible Limit Laws 775.3 The Link Between Hitting Times and Extreme Values 785.4 Uniformly Hyperbolic Systems 845.4.1 Gibbs Measures 855.4.2 First HTS Theorem 865.4.3 Markov Partitions 865.4.4 Two-Sided Shifts 885.4.5 Hyperbolic Diffeomorphisms 895.4.6 Additional Uniformly Hyperbolic Examples 905.5 Nonuniformly Hyperbolic Systems 915.5.1 Induced System 915.5.2 Intermittent Maps 925.5.3 Interval Maps with Critical Points 935.5.4 Higher Dimensional Examples of Nonuniform Hyperbolic Systems 945.6 Nonexponential Laws 956 Extreme Value Theory for Selected Dynamical Systems 976.1 Rare Events and Dynamical Systems 976.2 Introduction and Background on Extremes in Dynamical Systems 986.3 The Blocking Argument for Nonuniformly Expanding Systems 996.3.1 Assumptions on the Invariant Measure μ 996.3.2 Dynamical Assumptions on (f, χ, μ) 996.3.3 Assumption on the Observable Type 1006.3.4 Statement or Results 1016.3.5 The Blocking Argument in One Dimension 1026.3.6 Quantification of the Error Rates 1026.3.7 Proof of Theorem 6.3.1 1076.4 Nonuniformly Expanding Dynamical Systems 1086.4.1 Uniformly Expanding Maps 1086.4.2 Nonuniformly Expanding Quadratic Maps 1096.4.3 One-Dimensional Lorenz Maps 1106.4.4 Nonuniformly Expanding Intermittency Maps 1106.5 Nonuniformly Hyperbolic Systems 1136.5.1 Proof of Theorem 6.5.1 1156.6 Hyperbolic Dynamical Systems 1166.6.1 Arnold Cat Map 1166.6.2 Lozi-Like Maps 1186.6.3 Sinai Dispersing Billiards 1196.6.4 Hénon Maps 1196.7 Skew-Product Extensions of Dynamical Systems 1206.8 On the Rate of Convergence to an Extreme Value Distribution 1216.8.1 Error Rates for Specific Dynamical Systems 1236.9 Extreme Value Theory for Deterministic Flows 1266.9.1 Lifting to Xh 1296.9.2 The Normalization Constants 1296.9.3 The Lap Number 1306.9.4 Proof of Theorem 6.9.1 1316.10 Physical Observables and Extreme Value Theory 1336.10.1 Arnold Cat Map 1336.10.2 Uniformly Hyperbolic Attractors: The Solenoid Map 1376.11 Nonuniformly Hyperbolic Examples: the HÉNON and LOZI Maps 1406.12 Extreme Value Statistics for the Lorenz ’63 Model 1417 Extreme Value Theory for Randomly Perturbed Dynamical Systems 1457.1 Introduction 1457.2 Random Transformations via the Probabilistic Approach: Additive Noise 1467.2.1 Main Results 1497.3 Random Transformations via the Spectral Approach 1557.4 Random Transformations via the Probabilistic Approach: Randomly Applied Stochastic Perturbations 1597.5 Observational Noise 1637.6 Nonstationarity—the Sequential Case 1658 A Statistical Mechanical Point of View 1678.1 Choosing a Mathematical Framework 1678.2 Generalized Pareto Distributions for Observables of Dynamical Systems 1688.2.1 Distance Observables 1698.2.2 Physical Observables 1728.2.3 Derivation of the Generalized Pareto Distribution Parameters for the Extremes of a Physical Observable 1748.2.4 Comments 1768.2.5 Partial Dimensions along the Stable and Unstable Directions of the Flow 1778.2.6 Expressing the Shape Parameter in Terms of the GPD Moments and of the Invariant Measure of the System 1788.3 Impacts of Perturbations: Response Theory for Extremes 1808.3.1 Sensitivity of the Shape Parameter as Determined by the Changes in the Moments 1828.3.2 Sensitivity of the Shape Parameter as Determined by the Modification of the Geometry 1858.4 Remarks on the Geometry and the Symmetries of the Problem 1889 Extremes as Dynamical and Geometrical Indicators 1899.1 The Block Maxima Approach 1909.1.1 Extreme Value Laws and the Geometry of the Attractor 1919.1.2 Computation of the Normalizing Sequences 1929.1.3 Inference Procedures for the Block Maxima Approach 1949.2 The Peaks Over Threshold Approach 1969.2.1 Inference Procedures for the Peaks Over Threshold Approach 1969.3 Numerical Experiments: Maps Having Lebesgue Invariant Measure 1979.3.1 Maximum Likelihood versus L-Moment Estimators 2039.3.2 Block Maxima versus Peaks Over Threshold Methods 2049.4 Chaotic Maps With Singular Invariant Measures 2049.4.1 Normalizing Sequences 2059.4.2 Numerical Experiments 2089.5 Analysis of the Distance and Physical Observables for the HNON Map 2129.5.1 Remarks 2189.6 Extremes as Dynamical Indicators 2189.6.1 The Standard Map: Peaks Over Threshold Analysis 2199.6.2 The Standard Map: Block Maxima Analysis 2209.7 Extreme Value Laws for Stochastically Perturbed Systems 2239.7.1 Additive Noise 2259.7.2 Observational Noise 22910 Extremes as Physical Probes 23310.1 Surface Temperature Extremes 23310.1.1 Normal Rare and Extreme Recurrences 23510.1.2 Analysis of the Temperature Records 23510.2 Dynamical Properties of Physical Observables: Extremes at Tipping Points 23810.2.1 Extremes of Energy for the Plane Couette Flow 23910.2.2 Extremes for a Toy Model of Turbulence 24510.3 Concluding Remarks 24711 Conclusions 24911.1 Main Concepts of This Book 24911.2 Extremes Coarse Graining and Parametrizations 25311.3 Extremes of Nonautonomous Dynamical Systems 25511.3.1 A Note on Randomly Perturbed Dynamical Systems 25811.4 Quasi-Disconnected Attractors 26011.5 Clusters and Recurrence of Extremes 26111.6 Toward Spatial Extremes: Coupled Map Lattice Models 262Appendix A Codes 265A.1 Extremal Index 266A.2 Recurrences—Extreme Value Analysis 267A.3 Sample Program 271References 273Index 293

VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK.DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l’environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France.ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal.JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal.MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK.TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK.MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA.MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland.SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.

"This book presents a comprehensive approach to the theory of extreme events (extreme values) from the perspective of dynamical systems. It is nicely written by a group of authors from different parts of the globe".... "This book presents all the signs of becoming a significant tool in pure and applied mathematical research" Idris Assani on behalf of Mathematical Reviews, October 17

• 1 Introduction 11.1 A Transdisciplinary Research Area 11.2 Some Mathematical Ideas 41.3 Some Difficulties and Challenges in Studying Extremes 61.3.1 Finiteness of Data 61.3.2 Correlation and Clustering 81.3.3 Time Modulations and Noise 91.4 Extremes Observables and Dynamics 101.5 This Book 12Acknowledgments 142 A Framework for Rare Events in Stochastic Processes and Dynamical Systems 172.1 Introducing Rare Events 172.2 Extremal Order Statistics 192.3 Extremes and Dynamics 203 Classical Extreme Value Theory 233.1 The i.i.d. Setting and the Classical Results 243.1.1 Block Maxima and the Generalized Extreme Value Distribution 243.1.2 Examples 263.1.3 Peaks Over Threshold and the Generalized Pareto Distribution 283.2 Stationary Sequences and Dependence Conditions 293.2.1 The Blocking Argument 303.2.2 The Appearance of Clusters of Exceedances 313.3 Convergence of Point Processes of Rare Events 323.3.1 Definitions and Notation 333.3.2 Absence of Clusters 353.3.3 Presence of Clusters 353.4 Elements of Declustering 374 Emergence of Extreme Value Laws for Dynamical Systems 394.1 Extremes for General Stationary Processes—an Upgrade Motivated by Dynamics 404.1.1 Notation 414.1.2 The New Conditions 424.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 444.1.4 Proofs of Theorem 4.1.4 and Corollary 4.1.5 464.2 Extreme Values for Dynamically Defined Stochastic Processes 514.2.1 Observables and Corresponding Extreme Value Laws 534.2.2 Extreme Value Laws for Uniformly Expanding Systems 574.2.3 Example Revisited 594.2.4 Proof of the Dichotomy for Uniformly Expanding Maps 614.3 Point Processes of Rare Events 624.3.1 Absence of Clustering 624.3.2 Presence of Clustering 634.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes 654.4 Conditions Дq(un), D3(un) Dp(un)∗ and Decay of Correlations 664.5 Specific Dynamical Systems Where the Dichotomy Applies 704.5.1 Rychlik Systems 704.5.2 Piecewise Expanding Maps in Higher Dimensions 714.6 Extreme Value Laws for Physical Observables 725 Hitting and Return Time Statistics 755.1 Introduction to Hitting and Return Time Statistics 755.1.1 Definition of Hitting and Return Time Statistics 765.2 HTS Versus RTS and Possible Limit Laws 775.3 The Link Between Hitting Times and Extreme Values 785.4 Uniformly Hyperbolic Systems 845.4.1 Gibbs Measures 855.4.2 First HTS Theorem 865.4.3 Markov Partitions 865.4.4 Two-Sided Shifts 885.4.5 Hyperbolic Diffeomorphisms 895.4.6 Additional Uniformly Hyperbolic Examples 905.5 Nonuniformly Hyperbolic Systems 915.5.1 Induced System 915.5.2 Intermittent Maps 925.5.3 Interval Maps with Critical Points 935.5.4 Higher Dimensional Examples of Nonuniform Hyperbolic Systems 945.6 Nonexponential Laws 956 Extreme Value Theory for Selected Dynamical Systems 976.1 Rare Events and Dynamical Systems 976.2 Introduction and Background on Extremes in Dynamical Systems 986.3 The Blocking Argument for Nonuniformly Expanding Systems 996.3.1 Assumptions on the Invariant Measure μ 996.3.2 Dynamical Assumptions on (f, χ, μ) 996.3.3 Assumption on the Observable Type 1006.3.4 Statement or Results 1016.3.5 The Blocking Argument in One Dimension 1026.3.6 Quantification of the Error Rates 1026.3.7 Proof of Theorem 6.3.1 1076.4 Nonuniformly Expanding Dynamical Systems 1086.4.1 Uniformly Expanding Maps 1086.4.2 Nonuniformly Expanding Quadratic Maps 1096.4.3 One-Dimensional Lorenz Maps 1106.4.4 Nonuniformly Expanding Intermittency Maps 1106.5 Nonuniformly Hyperbolic Systems 1136.5.1 Proof of Theorem 6.5.1 1156.6 Hyperbolic Dynamical Systems 1166.6.1 Arnold Cat Map 1166.6.2 Lozi-Like Maps 1186.6.3 Sinai Dispersing Billiards 1196.6.4 Hénon Maps 1196.7 Skew-Product Extensions of Dynamical Systems 1206.8 On the Rate of Convergence to an Extreme Value Distribution 1216.8.1 Error Rates for Specific Dynamical Systems 1236.9 Extreme Value Theory for Deterministic Flows 1266.9.1 Lifting to Xh 1296.9.2 The Normalization Constants 1296.9.3 The Lap Number 1306.9.4 Proof of Theorem 6.9.1 1316.10 Physical Observables and Extreme Value Theory 1336.10.1 Arnold Cat Map 1336.10.2 Uniformly Hyperbolic Attractors: The Solenoid Map 1376.11 Nonuniformly Hyperbolic Examples: the HÉNON and LOZI Maps 1406.12 Extreme Value Statistics for the Lorenz ’63 Model 1417 Extreme Value Theory for Randomly Perturbed Dynamical Systems 1457.1 Introduction 1457.2 Random Transformations via the Probabilistic Approach: Additive Noise 1467.2.1 Main Results 1497.3 Random Transformations via the Spectral Approach 1557.4 Random Transformations via the Probabilistic Approach: Randomly Applied Stochastic Perturbations 1597.5 Observational Noise 1637.6 Nonstationarity—the Sequential Case 1658 A Statistical Mechanical Point of View 1678.1 Choosing a Mathematical Framework 1678.2 Generalized Pareto Distributions for Observables of Dynamical Systems 1688.2.1 Distance Observables 1698.2.2 Physical Observables 1728.2.3 Derivation of the Generalized Pareto Distribution Parameters for the Extremes of a Physical Observable 1748.2.4 Comments 1768.2.5 Partial Dimensions along the Stable and Unstable Directions of the Flow 1778.2.6 Expressing the Shape Parameter in Terms of the GPD Moments and of the Invariant Measure of the System 1788.3 Impacts of Perturbations: Response Theory for Extremes 1808.3.1 Sensitivity of the Shape Parameter as Determined by the Changes in the Moments 1828.3.2 Sensitivity of the Shape Parameter as Determined by the Modification of the Geometry 1858.4 Remarks on the Geometry and the Symmetries of the Problem 1889 Extremes as Dynamical and Geometrical Indicators 1899.1 The Block Maxima Approach 1909.1.1 Extreme Value Laws and the Geometry of the Attractor 1919.1.2 Computation of the Normalizing Sequences 1929.1.3 Inference Procedures for the Block Maxima Approach 1949.2 The Peaks Over Threshold Approach 1969.2.1 Inference Procedures for the Peaks Over Threshold Approach 1969.3 Numerical Experiments: Maps Having Lebesgue Invariant Measure 1979.3.1 Maximum Likelihood versus L-Moment Estimators 2039.3.2 Block Maxima versus Peaks Over Threshold Methods 2049.4 Chaotic Maps With Singular Invariant Measures 2049.4.1 Normalizing Sequences 2059.4.2 Numerical Experiments 2089.5 Analysis of the Distance and Physical Observables for the HNON Map 2129.5.1 Remarks 2189.6 Extremes as Dynamical Indicators 2189.6.1 The Standard Map: Peaks Over Threshold Analysis 2199.6.2 The Standard Map: Block Maxima Analysis 2209.7 Extreme Value Laws for Stochastically Perturbed Systems 2239.7.1 Additive Noise 2259.7.2 Observational Noise 22910 Extremes as Physical Probes 23310.1 Surface Temperature Extremes 23310.1.1 Normal Rare and Extreme Recurrences 23510.1.2 Analysis of the Temperature Records 23510.2 Dynamical Properties of Physical Observables: Extremes at Tipping Points 23810.2.1 Extremes of Energy for the Plane Couette Flow 23910.2.2 Extremes for a Toy Model of Turbulence 24510.3 Concluding Remarks 24711 Conclusions 24911.1 Main Concepts of This Book 24911.2 Extremes Coarse Graining and Parametrizations 25311.3 Extremes of Nonautonomous Dynamical Systems 25511.3.1 A Note on Randomly Perturbed Dynamical Systems 25811.4 Quasi-Disconnected Attractors 26011.5 Clusters and Recurrence of Extremes 26111.6 Toward Spatial Extremes: Coupled Map Lattice Models 262Appendix A Codes 265A.1 Extremal Index 266A.2 Recurrences—Extreme Value Analysis 267A.3 Sample Program 271References 273Index 293